Question

Grid In Find the measure of one exterior angle of a regular hexagon.

Short answer

60°

Step-by-step solution

Understanding the Regular Polygon

A hexagon is a polygon with 6 sides. A regular hexagon means all its sides and angles are equal. In a regular hexagon, we need to find the measure of one exterior angle.

Formula for Exterior Angle

The formula to find one exterior angle of a regular polygon is: \[\text{Exterior Angle} = \frac{360°}{n}\]where \(n\) is the number of sides of the polygon.

Substitute the Value of n

For a hexagon, the number of sides \(n = 6\). Substitute \(n\) into the formula:\[\text{Exterior Angle} = \frac{360°}{6}\]

Calculate the Exterior Angle

Divide 360 by 6 to find the measure of one exterior angle:\[\text{Exterior Angle} = 60°\]

Key concepts

Regular Hexagon

A regular hexagon is a fascinating geometric shape that consists of six sides, all of which are of equal length. This symmetry means that not only are all sides equal, but all the interior angles are equal as well. In a regular hexagon, each interior angle is 120 degrees. This uniformity is what distinguishes it as 'regular.' It’s important to note that this only applies to regular hexagons, and not all hexagons are regular. Hexagons are a common shape in the natural world, often seen in beehive structures and crystal formations.

When dealing with polygons like hexagons, understanding their properties helps in identifying solutions to geometry problems involving them. This is because these properties simplify calculations, like finding exterior angles and perimeters.

Hexagon Geometry

Hexagon geometry involves understanding the key attributes and properties of hexagons. A hexagon can be thought of as a six-sided polygon, and it can either be regular or irregular, depending on its angles and sides.

For instance, in a regular hexagon, all sides are equal, and all interior angles are also equal, each measuring 120 degrees. An interesting fact about the regular hexagon is that it can fit together perfectly on a plane without leaving any gaps, which is why it's a preferred shape for tiling, as seen in various forms of art and architecture.

The regularity and symmetry make them straightforward in calculations because their properties tend to yield predictable and simple-to-solve equations for area and angles.

Polygon Exterior Angles

Exterior angles play a crucial role in polygon geometry. The exterior angle of any polygon is the angle formed between one side of the polygon and the extension of an adjacent side. An essential property of polygon exterior angles is that the sum of all exterior angles in any polygon is always 360 degrees, regardless of the number of sides. This rule simplifies understanding and calculating exterior angles, especially in regular polygons.

To find the measure of an exterior angle in a regular polygon, like a regular hexagon, use the formula: \(\text{Exterior Angle} = \frac{360°}{n}\), where \(n\) is the number of sides. This relationship helps quickly determine angles, which is particularly handy when dealing with geometric designs, such as mosaics, or in engineering applications.

Geometry Formulas

Geometry formulas are vital tools for solving problems involving shapes, sizes, angles, and much more. For polygons, particularly regular ones, some of the most commonly used formulas include those for calculating interior and exterior angles.

Taking the example of a regular hexagon, the formula for finding an exterior angle simplifies as \(\text{Exterior Angle} = \frac{360°}{n}\). This basic formula can solve part of geometric problems efficiently, as seen with finding one exterior angle in a regular hexagon as 60 degrees.

Besides angles, formulas are also used for determining other attributes such as area and perimeter, improving understanding and design capabilities in various fields like engineering, architecture, and computer graphics. Mastery of these formulas enables students and professionals alike to handle geometric problems with confidence and precision.